Multinormality and Symmetry: Comparing Two Statistical Tests

Any multinormal distribution is symmetric. When we want to measure how far a distribution strays from axial symmetry, the Bowker test has long been the standard tool for the job. More recently, von Eye and Bogat (2004) and von Eye and Gardiner (2004) put forward a different approach. That test compares the frequencies actually observed across sectors of the multivariate space against the frequencies we would expect if the data were truly multinormal, with those expected values estimated under the assumption of multinormality.

Since the newer procedure works as an omnibus test of multinormality, there was good reason to think it would also pick up departures from axial symmetry. We set out to check that intuition through simulation. Four kinds of bivariate distributions went into the study: normal, uniform, inverse Laplace-transformed, and cube-root transformed.

The Bowker test behaved as anticipated. It identified the inverse Laplace-transformed distributions as the ones prone to breaking axial symmetry, while leaving the other three untouched. For symmetry violations and nothing else, in other words, Bowker proved a clean and targeted instrument.

The omnibus test told a broader story. It caught every single violation of axial symmetry, a perfect 100 percent sensitivity, but it did not stop there. Elevated skewness and kurtosis also tripped it, so both the uniform and the cube-root transformed distributions were flagged as departing from multinormality even though neither was asymmetric. The reach that makes the omnibus test useful as a general check on multinormality is the same reach that keeps it from isolating symmetry on its own. For that narrower question, the Bowker test remains the one that answers only what was asked.